The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions (another is Cantor dust). Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional plane, is homeomorphic to a subset of the Sierpinski carpet. For curves that cannot be drawn on a 2D surface without self-intersections, the corresponding universal curve is the Menger sponge, a higher-dimensional generalization.
The technique can be applied to repetitive tiling arrangement; triangle, square, hexagon being the simplest. It would seem impossible to apply it to other than rep-tile arrangements.
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The construction of the Sierpinski carpet begins with a square. The square is cut into 9 congruent subsquares in a 3-by-3 grid, and the central subsquare is removed. The same procedure is then applied recursively to the remaining 8 subsquares, ad infinitum. The Hausdorff dimension of the carpet is log 8/log 3 ≈ 1.8928.
The area of the carpet is zero (in standard Lebesgue measure).
The Sierpinski carpet can also be created by iterating every pixel in a square and using the following algorithm to decide if the pixel is filled. The following implementation is valid C, C++, and Java.
/** * Decides if a point at a specific location is filled or not. This works by iteration first checking if the pixel is unfilled in successively larger squares or cannot be in the center of any larger square. * @param x is the x coordinate of the point being checked with zero being the first pixel * @param y is the y coordinate of the point being checked with zero being the first pixel * @return 1 if it is to be filled or 0 if it is open */ int isSierpinskiCarpetPixelFilled(int x, int y) { while(x>0&&y>0) //when either of these reaches zero the pixel is determined to be on the edge at that square level and must be filled { if(x%3==1 && y%3==1) //checks if the pixel is in the center for the current square level return 0; x /= 3; //x and y are decremented to check the next larger square level y /= 3; } return 1; //if all possible square levels are checked and the pixel is not determined to be open it must be filled }
The topic of Brownian motion on the Sierpinski carpet has attracted interest in recent years.[1] Martin Barlow and Richard Bass have shown that a random walk on the Sierpinski carpet diffuses at a slower rate than an unrestricted random walk in the plane. The latter reaches a mean distance proportional to n1/2 after n steps, but the random walk on the discrete Sierpinski carpet reaches only a mean distance proportional to n1/β for some β > 2. They also showed that this random walk satisfies stronger large deviation inequalities (so called "sub-gaussian inequalities") and that it satisfies the elliptic Harnack inequality without satisfying the parabolic one. The existence of such an example was an open problem for many years.
Sierpiński, W.: Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée. C. r. hebd. Seanc. Acad. Sci., Paris 162, 629--632 (1916)